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G = C5×D21order 210 = 2·3·5·7

Direct product of C5 and D21

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×D21, C352S3, C153D7, C1053C2, C211C10, C7⋊(C5×S3), C3⋊(C5×D7), SmallGroup(210,9)

Series: Derived Chief Lower central Upper central

C1C21 — C5×D21
C1C7C21C105 — C5×D21
C21 — C5×D21
C1C5

Generators and relations for C5×D21
 G = < a,b,c | a5=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
7S3
21C10
3D7
7C5×S3
3C5×D7

Smallest permutation representation of C5×D21
On 105 points
Generators in S105
(1 97 83 56 42)(2 98 84 57 22)(3 99 64 58 23)(4 100 65 59 24)(5 101 66 60 25)(6 102 67 61 26)(7 103 68 62 27)(8 104 69 63 28)(9 105 70 43 29)(10 85 71 44 30)(11 86 72 45 31)(12 87 73 46 32)(13 88 74 47 33)(14 89 75 48 34)(15 90 76 49 35)(16 91 77 50 36)(17 92 78 51 37)(18 93 79 52 38)(19 94 80 53 39)(20 95 81 54 40)(21 96 82 55 41)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)(22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 42)(43 47)(44 46)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 84)(82 83)(85 87)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)

G:=sub<Sym(105)| (1,97,83,56,42)(2,98,84,57,22)(3,99,64,58,23)(4,100,65,59,24)(5,101,66,60,25)(6,102,67,61,26)(7,103,68,62,27)(8,104,69,63,28)(9,105,70,43,29)(10,85,71,44,30)(11,86,72,45,31)(12,87,73,46,32)(13,88,74,47,33)(14,89,75,48,34)(15,90,76,49,35)(16,91,77,50,36)(17,92,78,51,37)(18,93,79,52,38)(19,94,80,53,39)(20,95,81,54,40)(21,96,82,55,41), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,42)(43,47)(44,46)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,84)(82,83)(85,87)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)>;

G:=Group( (1,97,83,56,42)(2,98,84,57,22)(3,99,64,58,23)(4,100,65,59,24)(5,101,66,60,25)(6,102,67,61,26)(7,103,68,62,27)(8,104,69,63,28)(9,105,70,43,29)(10,85,71,44,30)(11,86,72,45,31)(12,87,73,46,32)(13,88,74,47,33)(14,89,75,48,34)(15,90,76,49,35)(16,91,77,50,36)(17,92,78,51,37)(18,93,79,52,38)(19,94,80,53,39)(20,95,81,54,40)(21,96,82,55,41), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,42)(43,47)(44,46)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,84)(82,83)(85,87)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97) );

G=PermutationGroup([(1,97,83,56,42),(2,98,84,57,22),(3,99,64,58,23),(4,100,65,59,24),(5,101,66,60,25),(6,102,67,61,26),(7,103,68,62,27),(8,104,69,63,28),(9,105,70,43,29),(10,85,71,44,30),(11,86,72,45,31),(12,87,73,46,32),(13,88,74,47,33),(14,89,75,48,34),(15,90,76,49,35),(16,91,77,50,36),(17,92,78,51,37),(18,93,79,52,38),(19,94,80,53,39),(20,95,81,54,40),(21,96,82,55,41)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21),(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,42),(43,47),(44,46),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,84),(82,83),(85,87),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97)])

C5×D21 is a maximal subgroup of   C5×S3×D7  D15⋊D7

60 conjugacy classes

class 1  2  3 5A5B5C5D7A7B7C10A10B10C10D15A15B15C15D21A···21F35A···35L105A···105X
order1235555777101010101515151521···2135···35105···105
size121211112222121212122222···22···22···2

60 irreducible representations

dim1111222222
type+++++
imageC1C2C5C10S3D7C5×S3D21C5×D7C5×D21
kernelC5×D21C105D21C21C35C15C7C5C3C1
# reps114413461224

Matrix representation of C5×D21 in GL2(𝔽41) generated by

160
016
,
4015
632
,
3227
359
G:=sub<GL(2,GF(41))| [16,0,0,16],[40,6,15,32],[32,35,27,9] >;

C5×D21 in GAP, Magma, Sage, TeX

C_5\times D_{21}
% in TeX

G:=Group("C5xD21");
// GroupNames label

G:=SmallGroup(210,9);
// by ID

G=gap.SmallGroup(210,9);
# by ID

G:=PCGroup([4,-2,-5,-3,-7,242,2883]);
// Polycyclic

G:=Group<a,b,c|a^5=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D21 in TeX

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